Disjunctive domination in graphs with minimum degree at least two

A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, ${\gamma }_2^d(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. In this paper, we show that if $G$ be a graph of order at least $3$, $\delta (G)\ge 2$ and with no component isomorphic to any of eight forbidden graphs, then ${\gamma }_2^d(G)\le \frac{\mathrm{\mid }G\mathrm{\mid }}{3}$. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that $G$ is a claw-free graph with minimum degree at least two.